Algebra

For the kind of algebraic structure, seeAlgebra over a field.For other uses, seeAlgebra (disambiguation).

Elementary algebrastudies which values solve equations formed using arithmetical operations.

Abstract algebrastudiesalgebraic structures,such as thering of integersgiven by the set ofintegers$(\mathbb {Z} )$together withoperationsofaddition($+$) andmultiplication($\times$).

Algebrais the branch ofmathematicsthat studies certain abstractsystems,known asalgebraic structures,and the manipulation of statements within those systems. It is a generalization ofarithmeticthat introducesvariablesandalgebraic operationsother than the standard arithmetic operations such asadditionandmultiplication.

Elementary algebrais the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables.Linear algebrais a closely related field that investigateslinear equationsand combinations of them calledsystems.It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions.

Abstract algebrastudies algebraic structures, which consist of asetofmathematical objectstogether with one or severaloperationsdefined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such asgroups,rings,andfields,based on the number of operations they use and thelaws they follow.Universal algebraandcategory theoryprovide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures.

Algebraic methods were first studied in theancient periodto solve specific problems in fields likegeometry.Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until the 16th and 17th centuries, when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond atheory of equationsto cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry,topology,number theory,andcalculus,and other fields of inquiry, likelogicand theempirical sciences.

Definition and etymology

Algebra is the branch of mathematics that studiesalgebraic structuresand theoperationsthey use.An algebraic structure is a non-emptysetofmathematical objects,such as theintegers,together with algebraic operations defined on that set, likeadditionandmultiplication.Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines the use ofvariablesinequationsand how to manipulate these equations.

Algebra is often understood as a generalization ofarithmetic.Arithmetic studies operations like addition,subtraction,multiplication, anddivision,in a particular domain of numbers, such as the real numbers.Elementary algebraconstitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in the form of variables in addition to numbers.A higher level of abstraction is found inabstract algebra,which is not limited to a particular domain and examines algebraic structures such asgroupsandrings.It extends beyond typical arithmetic operations by also covering other types of operations.Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates the characteristics of algebraic structures in general.

The wordalgebracomes from the title ofal-Khwarizmi's bookAl-Jabr.

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.When used as acountable noun,an algebra isa specific type of algebraic structurethat involves avector spaceequipped witha certain type of binary operation.Depending on the context, "algebra" can also refer to other algebraic structures, like aLie algebraor anassociative algebra.

The wordalgebracomes from theArabictermالجبر(al-jabr), which originally referred to the surgical treatment ofbonesetting.In the 9th century, the term received a mathematical meaning when the Persian mathematicianMuhammad ibn Musa al-Khwarizmiemployed it to describe a method of solving equations and used it in the title of a treatise on algebra,al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah[The Compendious Book on Calculation by Completion and Balancing] which was translated into Latin asLiber Algebrae et Almucabola.The word entered the English language in the 16th century fromItalian,Spanish,and medievalLatin.Initially, its meaning was restricted to thetheory of equations,that is, to the art of manipulatingpolynomial equationsin view of solving them. This changed in the 19th centurywhen the scope of algebra broadened to cover the study of diverse types of algebraic operations and structures together with their underlyingaxioms,the laws they follow.

Major branches

Elementary algebra

Main article:Elementary algebra

Algebraic expression notation:
1 – power (exponent)
2 – coefficient
3 – term
4 – operator
5 – constant term
$c$– constant
$x$$y$– variables

Elementary algebra, also called school algebra, college algebra, and classical algebra,is the oldest and most basic form of algebra. It is a generalization ofarithmeticthat relies onvariablesand examines how mathematicalstatementsmay be transformed.

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations ofaddition,subtraction,multiplication,division,exponentiation,extraction ofroots,andlogarithm.For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in$2+5=7$.

Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables aresymbolsfor unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, theequation$2\times 3=3\times 2$belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like thecommutative property of multiplication,which is expressed in the equation$a\times b=b\times a$.

Algebraic expressionsare formed by using arithmetic operations to combine variables and numbers. By convention, the lowercase letters$x$,$y$,and$z$represent variables. In some cases, subscripts are added to distinguish variables, as in$x_{1}$,$x_{2}$,and$x_{3}$.The lowercase letters$a$,$b$,and$c$are usually used forconstantsandcoefficients.The expression$5x+3$is an algebraic expression created by multiplying the number 5 with the variable$x$and adding the number 3 to the result. Other examples of algebraic expressions are$32xyz$and$64x_{1}^{2}+7x_{2}-c$.

Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is a statement formed by comparing two expressions, saying that they are equal. This can be expressed using theequals sign($=$), as in$5x^{2}+6x=3y+4$.Inequationsinvolve a different type of comparison, saying that the two sides are different. This can be expressed using symbols such as theless-than sign($<$), thegreater-than sign($>$), and the inequality sign ($\neq$). Unlike other expressions, statements can be true or false and their truth value usually depends on the values of the variables. For example, the statement$x^{2}=4$is true if$x$is either 2 or −2 and false otherwise.Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, such as the equation$2x+5x=7x$.Conditional equations are only true for some values. For example, the equation$x+4=9$is only true if$x$is 5.

The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known assolving the equationfor that variable. For example, the equation$x-7=4$can be solved for$x$by adding 7 to both sides, which isolates$x$on the left side and results in the equation$x=11$.

There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression$7x-3x$can be replaced with the expression$4x$since$7x-3x=(7-3)x=4x$by the distributive property.For statements with several variables,substitutionis a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that$y=3x$then one can simplify the expression$7xy$to arrive at$21x^{2}$.In a similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables.

Algebraic equations can be used to describe geometric figures. All values for$x$and$y$that solve the equation are interpreted as points. They are drawn as a red, upward-sloping line in the graph above.

Algebraic equations can be interpretedgeometricallyto describe spatial figures in the form of agraph.To do so, the different variables in the equation are understood ascoordinatesand the values that solve the equation are interpreted as points of a graph. For example, if$x$is set to zero in the equation$y=0.5x-1$,then$y$must be −1 for the equation to be true. This means that the$(x,y)$-pair$(0,-1)$is part of the graph of the equation. The$(x,y)$-pair$(0,7)$,by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of$(x,y)$-pairs that solve the equation.

Polynomials

Main article:Polynomial

A polynomial is an expression consisting of one or more terms that are added or subtracted from each other, like$x^{4}+3xy^{2}+5x^{3}-1$.Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive-integer power. A monomial is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials. Thedegree of a polynomialis the maximal value (among its terms) of the sum of the exponents of the variables (4 in the above example).Polynomials of degree one are calledlinear polynomials.Linear algebra studies systems of linear polynomials.A polynomial is said to beunivariateormultivariate,depending on whether it uses one or more variables.

Factorizationis used to rewrite an expression as a product of several factors. This technique is commonly used to determine the values of apolynomialthatevaluate to zero.For example, the polynomial$x^{2}-3x-10$can be factorized as$(x+2)(x-5)$.The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if$x$is either −2 or 5.Before the 19th century, much of algebra was devoted topolynomial equations,that isequationsobtained by equating a polynomial to zero. The first attempts for solving polynomial equations was to express the solutions in terms ofnth roots.The solution of a second-degree polynomial equation of the form$ax^{2}+bx+c=0$is given by thequadratic formula $$x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.$$ Solutions for the degrees 3 and 4 are given by thecubicandquarticformulas. There are no general solutions for higher degrees, as proven in the 19th century by the so-calledAbel–Ruffini theorem.Even when general solutions do not exist, approximate solutions can be found by numerical tools like theNewton–Raphson method.

Thefundamental theorem of algebraasserts that every univariate polynomial equation of positive degree withrealorcomplexcoefficients has at least one complex solution. Consequently, every polynomial of a positive degree can befactorizedinto linear polynomials. This theorem was proved at the beginning of the 19th century, but this does not close the problem since the theorem does not provide any way for computing the solutions.

Linear algebra

Main article:Linear algebra

Linear algebra employs the methods of elementary algebra to studysystems of linear equations.Anequation is linearif it can be expressed in the form$a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b$where$a_{1}$,$a_{2}$,...,$a_{n}$and$b$are constants. This means that no variables are multiplied with each other and no variables are raised to a power greater than one. For example, the equations$x_{1}-7x_{2}+3x_{3}=0$and$0.25x-y=4$are linear while the equations$x^{2}=y$and$3x_{1}x_{2}+15=0$arenon-linear.Asystem of linear equationsis a set of linear equations for which one is interested to the common solutions.Linear algebra is interested in manipulating and transforming systems of equations to solve them, which involves determining the values for which all equations are true at the same time.

Systems of linear equations are often expressed through tables of numbers known asmatricesandvectors,which represent the whole system in a single equation. This can be done by moving the variables to the left side of each equation and moving the constant terms to the right side. The system is then expressed by formulating a matrix that contains all the coefficients of the equations andmultiplyingit with thecolumn vectormade up of the variables.For example, the system of equations $${\begin{aligned}9x_{1}+3x_{2}-13x_{3}&=0\\2.3x_{1}+7x_{3}&=9\\-5x_{1}-17x_{2}&=-3\end{aligned}}$$ can be written as $${\begin{bmatrix}9&3&-13\\2.3&0&7\\-5&-17&0\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}0\\9\\-3\end{bmatrix}}.$$

Two central questions in linear algebra are whether a system of equations has any solutions and, if so, whether it has a unique solution. A system of equations has no solutions if it isinconsistent,meaning that two or more equations contradict each other. For example, the equations$x_{1}-3x_{2}=0$and$x_{1}-3x_{2}=7$contradict each other since no values of$x_{1}$and$x_{2}$exist that solve both equations at the same time. Only consistent systems of equations have solutions.Whether a consistent system of equations has a unique solution depends on the number of variables andindependent equations.Several equations are independent of each other if they do not provide the same information and cannot be derived from each other. A unique solution exists if the number of variables is the same as the number of independent equations.Underdetermined systems,by contrast, have more variables than independent equations and have an infinite number of solutions if they are consistent.

Methods of solving systems of linear equations range from the introductory, like substitutionand elimination,to more advanced techniques using algorithms based on matrix calculations, such asCramer's rule,theGauss–Jordan elimination,andLU decomposition.

Linear equations with two variables can be interpreted geometrically as lines. The solution of a system of linear equations is where the lines intersect.

Systems of equations can be interpreted as geometric figures. For systems with two variables, each equation represents alineintwo-dimensional space.The point where the two lines intersect is the solution of the full system because this is the only point that solves both the first and the second equation. For inconsistent systems, the two lines run parallel, meaning that there is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to seek solutions graphically by plotting the equations and determining where they intersect.The same principles also apply to systems of equations with more variables, with the difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond toplanesinthree-dimensional space,and the points where all planes intersect solve the system of equations.

Another approach defines linear algebra as the study oflinear mapsbetweenfinite-dimensionalvector spaces. A linear map is a function that transforms vectors from onevector spaceto another while preserving the operations ofvector additionandscalar multiplication.From this perspective, a matrix is a representation of a linear map: if one chooses a particularbasisto describe the vectors being transformed, then the entries in the matrix give the results of applying the linear map to the basis vectors.

Abstract algebra

Main article:Abstract algebra

Abstract algebra, also called modern algebra,studies different types ofalgebraic structures.An algebraic structure is a framework for understandingoperationsonmathematical objects,like the addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such asgroups,rings,andfields.The key difference between these types of algebraic structures lies in the number of operations they use and the laws they obey.Inmathematics education,abstract algebra refers to an advancedundergraduatecourse that mathematics majors take after completing courses in linear algebra.

Many algebraic structures rely on binary operations, which take two objects as their input and combine them into a single object as output, like addition and multiplication do.

On a formal level, an algebraic structure is asetof mathematical objects, called the underlying set, together with one or several operations.Abstract algebra is primarily interested inbinary operations,which take any two objects from the underlying set as inputs and map them to another object from this set as output.For example, the algebraic structure$\langle \mathbb {N},+\rangle$has thenatural numbers($\mathbb {N}$) as the underlying set and addition ($+$) as its binary operation.The underlying set can contain mathematical objects other than numbers and the operations are not restricted to regular arithmetic operations.For instance, the underlying set of thesymmetry groupof a geometric object is made up ofgeometric transformations,such asrotations,under which the object remainsunchanged.Its binary operation isfunction composition,which takes two transformations as input and has the transformation resulting from applying the first transformation followed by the second as its output.

Group theory

Main article:Group theory

Abstract algebra classifies algebraic structures based on the laws oraxiomsthat its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation isassociativeand has anidentity elementandinverse elements.An operation is associative if the order of several applications does not matter, i.e., if$(a\circ b)\circ c$is the same as$a\circ (b\circ c)$for all elements. An operation has an identity element or a neutral element if one elementeexists that does not change the value of any other element, i.e., if$a\circ e=e\circ a=a$.An operation has inverse elements if for any element$a$there exists a reciprocal element$a^{-1}$that undoes$a$.If an element operates on its inverse then the result is the neutral elemente,expressed formally as$a\circ a^{-1}=a^{-1}\circ a=e$.Every algebraic structure that fulfills these requirements is a group.For example,$\langle \mathbb {Z},+\rangle$is a group formed by the set ofintegerstogether with the operation of addition. The neutral element is 0 and the inverse element of any number$a$is$-a$.The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements.

Group theoryexamines the nature of groups, with basic theorems such as thefundamental theorem of finite abelian groupsand theFeit–Thompson theorem.The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a completeclassification of finite simple groups.

Ring theory and field theory

Main articles:Ring theoryandField (mathematics)

A ring is an algebraic structure with two operations ($\circ$and$\star$) that work similarly to addition and multiplication. All the requirements of groups also apply to the first operation: it is associative and has an identity element and inverse elements. Additionally, it is commutative, meaning that$a\circ b=b\circ a$is true for all elements. The axiom ofdistributivitygoverns how the two operations interact with each other. It states that$a\star (b\circ c)=(a\star b)\circ (a\star c)$and$(b\circ c)\star a=(b\star a)\circ (c\star a)$.Thering of integersis the ring denoted by$\langle \mathbb {Z},+,\times \rangle$.A ring becomes a field if both operations follow the axioms of associativity, commutativity, and distributivity and if both operations have an identity element and inverse elements.The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of$7$is${\tfrac {1}{7}}$,which is not part of the integers. Therational numbers,thereal numbers,and thecomplex numberseach form a field with the operations addition and multiplication.

Ring theoryis the study of rings, exploring concepts such assubrings,quotient rings,polynomial rings,andidealsas well as theorems such asHilbert's basis theorem.Field theory is concerned with fields, examiningfield extensions,algebraic closures,andfinite fields.Galois theoryexplores the relation between field theory and group theory, relying on thefundamental theorem of Galois theory.

Theories of interrelations among structures

Diagram of relations between some algebraic structures. For instance, its top right section shows that amagmabecomes asemigroupif its operation is associative.

Besides groups, rings, and fields, there are many other algebraic structures studied by algebra. They includemagmas,semigroups,monoids,abelian groups,commutative rings,modules,lattices,vector spaces,algebras over a field,andassociativeandnon-associative algebras.They differ from each other in regard to the types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements.For example, a magma becomes a semigroup if its operation is associative.

Homomorphismsare tools to examine structural features by comparing two algebraic structures.A homomorphism is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form$\langle A,\circ \rangle$and$\langle B,\star \rangle$then the function$h:A\to B$is a homomorphism if it fulfills the following requirement:$h(x\circ y)=h(x)\star h(y)$.The existence of a homomorphism reveals that the operation$\star$in the second algebraic structure plays the same role as the operation$\circ$does in the first algebraic structure.Isomorphismsare a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is abijectivehomomorphism, meaning that it establishes a one-to-one relationship between the elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure.

Subalgebrasrestrict their operations to a subset of the underlying set of the original algebraic structure.

Another tool of comparison is the relation between an algebraic structure and itssubalgebra.The algebraic structure and its subalgebra use the same operations,which follow the same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure.All operations in the subalgebra are required to beclosedin its underlying set, meaning that they only produce elements that belong to this set.For example, the set ofeven integerstogether with addition is a subalgebra of the full set of integers together with addition. This is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra because it is not closed: adding two odd numbers produces an even number, which is not part of the chosen subset.

Universal algebrais the study of algebraic structures in general. As part of its general perspective, it is not concerned with the specific elements that make up the underlying sets and considers operations with more than two inputs, such asternary operations.It provides a framework for investigating what structural features different algebraic structures have in common.One of those structural features concerns theidentitiesthat are true in different algebraic structures. In this context, an identity is auniversalequation or an equation that is true for all elements of the underlying set. For example, commutativity is a universal equation that states that$a\circ b$is identical to$b\circ a$for all elements.Avarietyis a class of all algebraic structures that satisfy certain identities. For example, if two algebraic structures satisfy commutativity then they are both part of the corresponding variety.

Category theoryexamines how mathematical objects are related to each other using the concept ofcategories.A category is a collection of objects together with a collection of so-calledmorphismsor "arrows" between those objects. These two collections must satisfy certain conditions. For example, morphisms can be joined, orcomposed:if there exists a morphism from object$a$to object$b$,and another morphism from object$b$to object$c$,then there must also exist one from object$a$to object$c$.Composition of morphisms is required to be associative, and there must be an "identity morphism" for every object.Categories are widely used in contemporary mathematics since they provide a unifying framework to describe and analyze many fundamental mathematical concepts. For example, sets can be described with thecategory of sets,and any group can be regarded as the morphisms of a category with just one object.

History

Main articles:History of algebraandTimeline of algebra

TheRhind Mathematical Papyrusfromancient Egypt,datedc. 1650 BCE,is one of the earliest documents discussing algebraic problems.

The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities. These developments happened in the ancient period inBabylonia,Egypt,Greece,China,andIndia.One of the earliest documents on algebraic problems is theRhind Mathematical Papyrusfrom ancient Egypt, which was written around 1650 BCE.It discusses solutions tolinear equations,as expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?" Babylonian clay tablets from around the same time explain methods to solve linear andquadratic polynomial equations,such as the method ofcompleting the square.

Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main interest was geometry rather than algebra, but they employed algebraic methods to solve geometric problems. For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified inPythagoras' formulation of thedifference of two squaresmethod and later inEuclid'sElements.In the 3rd century CE,Diophantusprovided a detailed treatment of how to solve algebraic equations in a series of books calledArithmetica.He was the first to experiment with symbolic notation to express polynomials.In ancient China,The Nine Chapters on the Mathematical Art,a book composed over the period spanning from the 10th century BCE to the 2nd century CE,explored various techniques for solving algebraic equations, including the use of matrix-like constructs.

There is no unanimity as to whether these early developments are part of algebra or only precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications.This changed with the Persian mathematicianal-Khwarizmi,who published hisThe Compendious Book on Calculation by Completion and Balancingin 825 CE. It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides.Other influential contributions to algebra came from the Arab mathematicianThābit ibn Qurraalso in the 9th century and the Persian mathematicianOmar Khayyamin the 11th and 12th centuries.

In India,Brahmaguptainvestigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his innovations were the use of zero and negative numbers in algebraic equations.The Indian mathematiciansMahāvīrain the 9th century andBhāskara IIin the 12th century further refined Brahmagupta's methods and concepts.In 1247, the Chinese mathematicianQin Jiushaowrote theMathematical Treatise in Nine Sections,which includesan algorithmfor thenumerical evaluation of polynomials,including polynomials of higher degrees.

François Viète(left) andRené Descartesinvented a symbolic notation to express equations in an abstract and concise manner.

The Italian mathematicianFibonaccibrought al-Khwarizmi's ideas and techniques to Europe in books including hisLiber Abaci.In 1545, the Italian polymathGerolamo Cardanopublished his bookArs Magna,which covered many topics in algebra, discussedimaginary numbers,and was the first to present general methods for solvingcubicandquartic equations.In the 16th and 17th centuries, the French mathematiciansFrançois VièteandRené Descartesintroduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner. Their predecessors had relied on verbal descriptions of problems and solutions.Some historians see this development as a key turning point in the history of algebra and consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation.

Garrett Birkhoffdeveloped many of the foundational concepts of universal algebra.

In the 17th and 18th centuries, many attempts were made to find general solutions to polynomials of degree five and higher. All of them failed.At the end of the 18th century, the German mathematicianCarl Friedrich Gaussproved thefundamental theorem of algebra,which describes the existence ofzerosof polynomials of any degree without providing a general solution.At the beginning of the 19th century, the Italian mathematicianPaolo Ruffiniand the Norwegian mathematicianNiels Henrik Abelwereable to showthat no general solution exists for polynomials of degree five and higher.In response to and shortly after their findings, the French mathematicianÉvariste Galoisdeveloped what came later to be known asGalois theory,which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation ofgroup theory.Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory.

Starting in the mid-19th century, interest in algebra shifted from the study of polynomials associated with elementary algebra towards a more general inquiry into algebraic structures, marking the emergence ofabstract algebra.This approach explored the axiomatic basis of arbitrary algebraic operations.The invention of new algebraic systems based on different operations and elements accompanied this development, such asBoolean algebra,vector algebra,andmatrix algebra.Influential early developments in abstract algebra were made by the German mathematiciansDavid Hilbert,Ernst Steinitz,andEmmy Noetheras well as the Austrian mathematicianEmil Artin.They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, like groups, rings, and fields.

The idea of the even more general approach associated with universal algebra was conceived by the English mathematicianAlfred North Whiteheadin his 1898 bookA Treatise on Universal Algebra.Starting in the 1930s, the American mathematicianGarrett Birkhoffexpanded these ideas and developed many of the foundational concepts of this field.The invention of universal algebra led to the emergence of various new areas focused on the algebraization of mathematics—that is, the application of algebraic methods to other branches of mathematics. Topological algebra arose in the early 20th century, studying algebraic structures such astopological groupsandLie groups.In the 1940s and 50s,homological algebraemerged, employing algebraic techniques to studyhomology.Around the same time,category theorywas developed and has since played a key role in thefoundations of mathematics.Other developments were the formulation ofmodel theoryand the study offree algebras.

Applications

See also:Applied mathematics

The influence of algebra is wide-reaching, both within mathematics and in its applications to other fields.The algebraization of mathematics is the process of applying algebraic methods and principles to otherbranches of mathematics,such asgeometry,topology,number theory,andcalculus.It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other.

The algebraic equation$x^{2}+y^{2}+z^{2}=1$describes asphereat theoriginwith a radius of 1.

One application, found in geometry, is the use of algebraic statements to describe geometric figures. For example, the equation$y=3x-7$describes a line in two-dimensional space while the equation$x^{2}+y^{2}+z^{2}=1$corresponds to aspherein three-dimensional space. Of special interest toalgebraic geometryarealgebraic varieties,which are solutions tosystems of polynomial equationsthat can be used to describe more complex geometric figures.Algebraic reasoning can also solve geometric problems. For example, one can determine whether and where the line described by$y=x+1$intersects with the circle described by$x^{2}+y^{2}=25$by solving the system of equations made up of these two equations.Topology studies the properties of geometric figures ortopological spacesthat are preserved under operations ofcontinuous deformation.Algebraic topologyrelies on algebraic theories such asgroup theoryto classify topological spaces. For example,homotopy groupsclassify topological spaces based on the existence ofloopsorholesin them.

Number theory is concerned with the properties of and relations between integers.Algebraic number theoryapplies algebraic methods and principles to this field of inquiry. Examples are the use of algebraic expressions to describe general laws, likeFermat's Last Theorem,and of algebraic structures to analyze the behavior of numbers, such as thering of integers.The related field ofcombinatoricsutilizes algebraic techniques to solve problems related to counting, arrangement, and combination of discrete objects. An example inalgebraic combinatoricsis the application of group theory to analyzegraphsand symmetries.The insights of algebra are also relevant to calculus, which utilizes mathematical expressions to examinerates of changeandaccumulation.It relies on algebra, for instance, to understand how these expressions can be transformed and what role variables play in them.Algebraic logicemploys the methods of algebra to describe and analyze the structures and patterns that underlielogical reasoning,exploring both the relevant mathematical structures themselves and their application to concrete problems of logic.It includes the study ofBoolean algebrato describepropositional logicas well as the formulation and analysis of algebraic structures corresponding to more complexsystems of logic.

The faces of aRubik's cubecan be rotated to change the arrangement of colored patches. The resulting permutations form a group called theRubik's Cube group.

Algebraic methods are also commonly employed in other areas, like the natural sciences. For example, they are used to expressscientific lawsand solve equations inphysics,chemistry,andbiology.Similar applications are found in fields likeeconomics,geography,engineering(includingelectronicsandrobotics), andcomputer scienceto express relationships, solve problems, and model systems.Linear algebra plays a central role inartificial intelligenceandmachine learning,for instance, by enabling the efficient processing and analysis of largedatasets.Various fields rely on algebraic structures investigated by abstract algebra. For example, physical sciences likecrystallographyandquantum mechanicsmake extensive use of group theory,which is also employed to study puzzles such asSudokuandRubik's cubes,andorigami.Bothcoding theoryandcryptologyrely on abstract algebra to solve problems associated withdata transmission,like avoiding the effects ofnoiseand ensuringdata security.

Education

See also:Mathematics education

Balance scalesare used in algebra education to help students understand how equations can be transformed to determine unknown values.

Algebra education mostly focuses on elementary algebra, which is one of the reasons why elementary algebra is also called school algebra. It is usually not introduced untilsecondary educationsince it requires mastery of the fundamentals of arithmetic while posing new cognitive challenges associated with abstract reasoning and generalization.It aims to familiarize students with the formal side of mathematics by helping them understand mathematical symbolism, for example, how variables can be used to represent unknown quantities. An additional difficulty for students lies in the fact that, unlike arithmetic calculations, algebraic expressions are often difficult to solve directly. Instead, students need to learn how to transform them according to certain laws, often with the goal of determining an unknown quantity.

Some tools to introduce students to the abstract side of algebra rely on concrete models and visualizations of equations, including geometric analogies, manipulatives including sticks or cups, and "function machines" representing equations asflow diagrams.One method usesbalance scalesas a pictorial approach to help students grasp basic problems of algebra. The mass of some objects on the scale is unknown and represents variables. Solving an equation corresponds to adding and removing objects on both sides in such a way that the sides stay in balance until the only object remaining on one side is the object of unknown mass.Word problemsare another tool to show how algebra is applied to real-life situations. For example, students may be presented with a situation in which Naomi's brother has twice as many apples as Naomi. Given that both together have twelve apples, students are then asked to find an algebraic equation that describes this situation ($2x+x=12$) and to determine how many apples Naomi has($x=4$).

At the university level, mathematics students encounter advanced algebra topics from linear and abstract algebra. Initialundergraduatecourses in linear algebra focus on matrices, vector spaces, and linear maps. Upon completing them, students are usually introduced to abstract algebra, where they learn about algebraic structures like groups, rings, and fields, as well as the relations between them. The curriculum typically also covers specific instances of algebraic structures, such as the systems of the rational numbers, the real numbers, and the polynomials.

See also

• Algebra over a set
• Algebra tile
• Algebraic combinatorics
• Bézout's theorem
• C*-algebra
• Clifford algebra
• Commutative algebra
• Composition algebra
• Computer algebra
• Cyclotomic polynomial
• Diophantine equation
• Discrete Fourier transform
• Discrete group
• Dual space
• Eigenvalues and eigenvectors
• Equivalence class
• Equivalence relation
• Exterior algebra
• F-algebra
• F-coalgebra
• Finite field
• Fundamental theorem of finitely generated abelian groups
• Gaussian elimination
• Geometric algebra
• Gröbner basis
• Heyting algebra
• Hilbert space
• Hilbert's Nullstellensatz
• Hilbert's syzygy theorem
• Hopf algebra
• Lattice (group)
• Lie group
• Linear form
• Linear subspace
• Matrix decomposition
• Multilinear map
• Non-associative algebra
• Outline of algebra
• Quaternion
• Rational function
• Relational algebra
• Representation theory
• Root of unity
• Scheme theory
• Sigma-algebra
• Singular value decomposition
• Spectral theory
• Symmetric algebra
• T-algebra
• Tensor
• Tensor algebra
• Wiles' proof of Fermat's Last Theorem

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